Edwin Jonk: Yes it can. That is, if you sell a product/service that is unique and cannot be bought elsewhere or there aren`t any substitutes for your product. Then you will rank for the specific product/service. For example, think about events. If you are the organizer of a particular event in a particular place on a particular time then you will rank for it (assuming that everything is crawlable and optimized).
That said (and following the event example), you probably want to reach out to local newspaper which might publish an article on their site or contact the local radio-station for a live interview, which they might publish on their site. All those "normal" marketing things can lead to links or other off-site signals.
So if you are doing it right there shouldn`t be a question like "should we do off-site or on-site SEO?". In contrast, you should take every opportunity that is presented to you. Maybe you only do the on-page SEO, but that doesn`t mean the company as a whole should only do on-page SEO. Furthermore you can use the other marketing efforts on your site. For example, when the local newspaper publish an article about the event you can add a "featured on" section on your page/site.
Selected answers from the Dumb SEO Questions G+ community.
Gareth Lisle: A combination of SEO with a well thought out landing page and you're on to a real winner! ;But can an optimised landing page survive do well without good off-site SEO too?
Joseph Paulino: As far as I've experienced and tested time and time again that good off-page SEO is definitely necessary. Being able to ;implement everything you know in on-page SEO isn't always enough.
You can continue increasing the number of search engine optimization guidelines to implement as given by major search engines but you'll find that trying to perfect on-page SEO isn't a guarantee that you'll beat your competitors in the SERPs.